Abstract
The cosmological particle horizon is the maximum measurable length in the Universe. The existence of such a maximum observable length scale implies a modification of the quantum uncertainty principle. Thus due to non-locality of quantum mechanics, the global properties of the Universe could produce a signature on the behaviour of local quantum systems. A Generalized Uncertainty Principle (GUP) that is consistent with the existence of such a maximum observable length scale l_max is Δ x Δ p ≥ ℏ/21/1-α Δ x² where α = l_max⁻²≃ (H₀/c)² (H₀ is the Hubble parameter and c is the speed of light). In addition to the existence of a maximum measurable length l_max=1/√α, this form of GUP implies also the existence of a minimum measurable momentum p_min=3 √3/4ℏ √α. Using appropriate representation of the position and momentum quantum operators we show that the spectrum of the one dimensional harmonic oscillator becomes E_n=2n+1+λ_n α where E_n≡ 2E_n/ℏ ω is the dimensionless properly normalized n^th energy level, α is a dimensionless parameter with α≡ α ℏ/m ω and λ_n∼ n² for n≫ 1 (we show the full form of λ_n in the text). For a typical vibrating diatomic molecule and l_max=c/H₀ we find α∼ 10⁻⁷⁷ and therefore for such a system, this effect is beyond reach of current experiments. However, this effect could be more important in the early universe and could produce signatures in the primordial perturbation spectrum induced by quantum fluctuations of the inflaton field.